Designing Bézier conic segments with monotone curvature

Abstract

For some applications of computer-aided geometric design it is important to maintain strictly monotone curvature along a curve segment. Here we analyze the curvature distributions of segments of conic sections represented as rational quadratic Bézier curves in standard form. We show that if the end points and the weight are fixed, then the curvature of the conic segment will be strictly monotone if and only if the other control point lies inside well-defined regions bounded by circular arcs. We also show that if the turning angle of the curve is less than or equal to 90°, then there are always values of the weight that ensure strict monotonicity of the curvature distribution. Furthermore, bounds on such values of the weight are easily computed.